PERFORMANCE ASSESSMENT OF PID

CONTROLLERS∗

W. Tan†, H. J. Marquez‡, and T. Chen§

Abstract: Criteria based on disturbance rejection and system robustness are proposed to assess

the performance of PID controllers. The robustness is measured by a two-block structured singu-

lar value, and the disturbance rejection is measured by the minimum singular value of the integral

gain matrix. Examples show that the criteria can be applied to a variety of processes, whether

they are stable, integrating or unstable; single-loop or multi-loop.

Key Words: PID Control; Tuning; Performance; Robustness; Structured Singular Value.

1 Introduction

PID controllers are widely used in the industry due to their simplicity and ease of re-tuning on-line

[1]. In the past four decades there are numerous papers dealing with the tuning of PID controllers.

∗Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, SEM,and by the Research Foundation for the Doctoral Program, NCEPU

†Department of Automation, North China Electric Power University, Zhuxinzhuang, Dewai, Beijing 102206,China. E-mail: wtan@ieee.org

‡Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada.E-mail: marquez@ece.ualberta.ca

§Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada.E-mail: tchen@ece.ualberta.ca

1

See, for example, references [2–9] for stable processes; references [10–17] for integrating and

unstable processes; and references [18–23] for multivariable processes.

A natural question arises: How can the PID settings obtained by different methods be com-

pared? Or more generally, how can the performance of a controller be assessed? In process

control, minimum variance has been used as a criterion for assessing closed-loop performance

for decades [24, 25]. This criterion is a valuable measure of system performance but it pays lit-

tle attention to the traditional performance such as setpoint tracking and disturbance rejection.

Besides, another important factor of system performance – robustness is not addressed directly.

Clearly a criterion that can be used for stable, integrating or unstable; single- and/or multi-loop

processes would be highly desirable. This criterion should include time domain property as well

as frequency domain robustness specification.

For single-loop processes, the integral error is a good measure of system performance and the

gain-phase margin is a good robustness measure. Thus, a combination of these two elements can

serve as a criterion for system performance assessment. A comparison of the gain-phase margins

of some well-known PID tuning methods has been reported in [26]. But unfortunately gain and

phase margins are not suitable for multiloop processes.

In this paper we will propose criteria to assess system performance. The criteria reflect distur-

bance rejection performance and system robustness. The robustness is measured by a two-block

structured singular value, and the disturbance rejection is measured by the minimum singular

value of the integral gain matrix. Examples show that the criteria can be applied to a variety of

processes, whether they are stable, integrating or unstable; single-loop or multi-loop.

2

2 Performance Assessment of Closed-loop Systems

It is well-known that a well-designed control system should meet the following requirements

besides nominal stability:

• Disturbance attenuation

• Setpoint tracking

• Robust stability and/or robust performance

The first two requirements are traditionally referred to as ‘performance’ and the third, ‘robust-

ness’ of a control system.

2.1 Performance

The integral error is a good measure for evaluating the setpoint and disturbance response. The

followings are some commonly used criteria based on the integral error for a step setpoint or

disturbance response:

IAE :∫ ∞

0 |e(t)|dt

ITAE :∫ ∞

0 t|e(t)|dt

ISE :∫ ∞

0 e(t)2dt

ITSE :∫ ∞

0 te(t)2dt

ISTE :∫ ∞

0 t2e(t)2dt

These criteria, however, are not suitable for multivariable processes, since each criterion is de-

fined for a single-loop process.

3

Consider the unity feedback system (possibly multi-loop) shown in Fig. 1. Since disturbance

rejection is more common in industrial processes than setpoint tracking, the performance of the

system may be evaluated by its ability to reject disturbance.

K Gr y

_

Gd

e

d

+

Figure 1: Typical unity feedback configuration

The transfer function from d to y is

Tyd = (I +GK)−1Gd (1)

Assume our controller K has integral action, we can decompose it as

K(s) = Ki/s+Km(s) (2)

where Ki is the integral gain and Km is the part of the controller without integral action. Then at

low frequency, we have

σ((I +GK)−1Gd)( jω)) ≤ | jω|σ((G( jω)Ki)−1Gd( jω))

≤ | jω| σ(Gd( jω))σ(G( jω))σ(Ki)

(3)

4

where σ(·) and σ(·) denote the maximum and minimum singular value of a matrix, respectively.

In industrial processes, the disturbance usually occurs at low frequency, so to reject a disturbance,

the most important element of a controller is its integral gain, or specifically, the minimum sin-

gular value of the integral gain, thus it can serve as a measure of system performance.

As pointed out in [1] for a single-loop process,

∫ ∞

0e(t)dt =

1Ki

so the integral gain is related to the integral of the error (IE). Moreover, if the response is critically

damped, IE would be equal to IAE. So the minimum singular value of the integral gain is a natural

extension as a performance measure to multi-loop processes.

2.2 Robustness

For robust stability, a common choice of representing uncertainty for a multivariable system is

the multiplicative perturbation, and the maximum singular value of the complementary sensitivity

matrix is a measure of robustness against this kind of uncertainty, which is usually frequency-

dependent, and suited for the unmodeled dynamics instead of parameter variations. The coprime

factor uncertainty can represent model uncertainty in a better way [27]. The uncertain model is

represented as:

G∆ = (M +∆M)−1(N +∆N) (4)

where G = M−1N is a left normalized coprime factorization of the nominal plant model, and the

uncertainty structure is

∆ = [ ∆M ∆N ], ‖∆‖∞ < γ (5)

5

Then the system is robustly stable if and only if

ε :=

∥∥∥∥∥∥∥

⎡⎢⎣ I

K

⎤⎥⎦(I +GK)−1M−1

∥∥∥∥∥∥∥∞

≤ 1γ

(6)

So ε can serve as a measure of system robustness.

However, we note that this uncertainty clearly ignores the structure of ∆M and ∆N . Suppose

∆M = W1∆1,∆N = W2∆2 (7)

and define

∆ =

⎡⎢⎣ ∆1 0

0 ∆2

⎤⎥⎦ (8)

then

∆ = [ W1 W2 ]∆ (9)

For this uncertainty structure, we have

(I +G∆K)−1

=

⎛⎜⎝I +

[W1 W2

]⎡⎢⎣ ∆1 0

0 ∆2

⎤⎥⎦

⎡⎢⎣ I

K

⎤⎥⎦(I +GK)−1M−1

⎞⎟⎠

−1

·

(I +GK)−1(1+ M−1W1∆1) (10)

By the definition of structured singular values [28], the closed-loop system is robustly stable for

6

all ‖∆‖∞ < γ if and only if

µ∆

⎛⎜⎝

⎡⎢⎣ I

K

⎤⎥⎦(I +GK)−1M−1[ W1 W2 ]

⎞⎟⎠ <

1γ

(11)

If we choose special weightings as follows:

W1 = M;W2 = N (12)

and define

εm := µ∆

⎛⎜⎝

⎡⎢⎣ I

K

⎤⎥⎦(I +GK)−1[ I G ]

⎞⎟⎠ (13)

Then εm is a better measure of system robustness. We note that now the class of uncertain plants

can be represented as

G∆ = (M + M∆1)−1(N + N∆2) = (I +∆1)−1G(I +∆2) (14)

so it can represent simultaneous input multiplicative and inverse output multiplicative uncertainty.

If we treat a disturbance as a model uncertainty, then it can also represent simultaneous input and

output disturbance.

For a single-loop system, it can be shown that

εm = maxω

(|S( jω)|+ |T ( jω)|) (15)

where S and T are the sensitivity and complementary sensitivity functions of the closed-loop

system, respectively. The value approaches 1 at low and high frequencies and the maximum

7

occurs at the mid-range frequencies. Compared with the usual indicator such as Ms, the peak of

the sensitivity function, or Mp, the peak of the complementary sensitivity function, the measure

is more appropriate since it bounds both Ms and Mp simultaneously.

In summary we can assess the performance of a controller by evaluating the minimum sin-

gular value of its integral gain matrix, and assess the robustness by the robustness measure εm

defined in (13). We mainly concern with the disturbance response. The setpoint response can

always be improved by using a setpoint filter or a setpoint weighting.

The discussion above suggests that we can design an ‘optimal’ PID controller by solving the

following optimization problem:

maxσ(Ki) (16)

under the constraint

µ∆

⎛⎜⎝

⎡⎢⎣ I

K

⎤⎥⎦(I +GK)−1[ I G ]

⎞⎟⎠ < γm (17)

where γm is a given robust stability requirement. The problem amounts to maximizing the integral

action under the constraint of a certain degree of robust stability, a generalization of the idea used

in [29, 30] for single-loop processes.

The problem proposed is a nonconvex optimization problem thus it is not easy to solve di-

rectly. However, the loop-shaping H∞ approach provides a solution to a suboptimal problem.

Details can be found in [31].

3 Illustrative Examples

In this section, we will apply the criteria proposed in the previous section to analyze the PID

controller settings for some typical processes.

8

3.1 A first-order plus deadtime (FOPDT) process

Consider a process with the following model:

P(s) =1

s+1e−0.5s (18)

Table 1 shows the PID settings tuned by the following well-known tuning rules:

1) Ziegler-Nichols (Z-N) [2].

2) Cohen-Coon (C-C) [3].

3) Internal model control (IMC) [5]. The IMC method has a tuning parameter. The smaller it

is, the better performance the closed-loop system will have, and the less robust the closed-

loop system is. Here the tuning parameter λ is chosen as 0.25 of the delay, the smallest

value as suggested by [5].

4) Gain-phase margin (GPM) [4, 8]. Since different pair of gain-phase margin will result in

different PID settings, here we choose the tuning formula given in [7] where the gain-phase

margin is optimized.

5) Optimum integral error for load disturbance (ISE-load, ISTE-load, ITAE-load) [7, 32].

6) Optimum integral error for setpoint change (ISE-setpoint, ISTE-setpoint, ITAE-setpoint)

[7, 32].

It can be observed that the resulting controllers can be divided into three groups:

i) Controllers tuned by IMC, GPM, ISTE-setpoint and IAE-setpoint methods have small in-

tegral gains and small robustness measures.

9

Table 1: PID settings for example 1

Kp Ti Td Ki εm

IMC 2 1.25 0.2 1.6 3.193GPM 1.938 1.164 0.208 1.665 3.092

ISE-setpoint 1.952 0.989 0.264 1.973 3.646ISTE-setpoint 1.940 1.152 0.206 1.684 3.078IAE-setpoint 1.674 1.204 0.192 1.390 2.544

ITAE-setpoint 2.393 1.201 0.175 1.992 4.054Z-N 2.285 0.855 0.214 2.617 3.912C-C 2.917 1.209 0.167 2.833 6.024

ISE-load 2.885 0.532 0.285 5.422 9.852ISTE-load 2.876 0.642 0.231 4.477 6.698IAE-load 2.673 0.852 0.20 3.139 5.188

ITAE-load 2.475 0.813 0.183 3.045 4.258

ii) Controllers tuned by Z-N, ITAE-setpoint, ITAE-load and ISE-setpoint methods have medium

integral gains and medium robustness measures.

iii) Controllers tuned by C-C, ISE-load, ISTE-load and IAE-load methods have large integral

gains and large robustness measures.

Fig. 2 shows the the closed-loop system responses of all the PID controllers for a step setpoint

change of magnitude 1 at t = 0 following a step load disturbance of magnitude 1 at t = 10 for the

nominal model and for the perturbed case that the deadtime increases by 20%. It is observed that

the integral gains and robustness measures given by the first group are too small, thus the closed-

loop systems are very robust but the load rejection performance can be further improved. The

integral gains and robustness measures given by the third group are too large, thus the closed-loop

systems show oscillatory responses and are not robust. The second group gives proper integral

gains and robustness measures.

10

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Pro

cess

Out

put

IMCG−PISTE−setpointIAE−setpoint

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Pro

cess

Out

put

IMCG−PISTE−setpointIAE−setpoint

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

Pro

cess

Out

put

Z−NITAE−setpointITAE−loadISE−setpoint

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time

Pro

cess

Out

put

Z−NITAE−setpointITAE−loadISE−setpoint

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time

Pro

cess

Out

put

C−CISE−loadISTE−loadIAE−load

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

Time

Pro

cess

Out

put

C−CISE−loadISTE−loadIAE−load

(a) Nominal model (b) Deadtime increases by 20%

Figure 2: Responses of different PID settings for example 1

11

3.2 A process with complex poles

Second-order plus deadtime processes (SOPDT) are harder to tune than FOPDT processes due to

the existence of (possibly) underdamped complex poles. Nonetheless, the criteria apply also to

such processes. To illustrate, consider a process with the following model:

P(s) =1

s2 +0.1s+1e−0.2s (19)

It has a very small damping ratio, thus represents a heavily oscillatory process.

Table 2: PID settings for example 2

Kp Ti Td Ki εm

γm = 3 1.996 1.419 0.985 1.406 2.942γm = 4 3.825 1.550 0.710 2.468 3.920γm = 5 5.365 1.575 0.608 3.405 4.970

Table 2 shows the PID settings tuned by solving the (sub)optimization problem proposed at

the end of the previous section with different values of γm. We do not claim that the solutions are

optimal. The suboptimal solutions are just used to illustrate the impact of the robustness measure

on system performance.

Fig. 2 shows the the closed-loop system responses of all the PID controllers for a step setpoint

change of magnitude 1 at t = 0 following a step load disturbance of magnitude 1 at t = 20 for the

nominal model and for the perturbed case that the deadtime increases by 30%. It is observed that

the integral gain and robustness measure computed with γm = 3 are too small, thus the closed-

loop systems are very robust but the load rejection performance can be further improved. The

integral gain and robustness measure computed with γm = 5 are too large, thus the response of

the closed-loop system is oscillatory and not robust. γm = 4 gives a good trade-off.

Extensive simulations show that for a stable process the robustness measure εm defined in (13)

12

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Pro

cess

Out

put

0 5 10 15 20 25 30 35 400

0.5

1

1.5

Time

Pro

cess

Out

put

(a) Nominal (b) Deadtime increases by 30%

Figure 3: Responses of different PID settings for example 2(dashed: γm = 3; solid: γm = 4; dashdotted: γm = 5)

should lie between 3 and 5 to have a good compromise between performance and robustness.

3.3 A first-order delayed unstable process (FODUP)

PID tuning for integrating and unstable processes is much harder than that for stable processes.

There are few simple tuning formulas as those in the case of stable processes available in the

literature. Here we consider a first-order delayed unstable process as an illustrating example:

P(s) =1

s−1e−0.4s (20)

Table 3 shows the PID settings tuned by typical methods found in the literature, and the PID

setting tuned by solving the optimization problem proposed at the end of the previous section

with γm = 5. Also shown are the corresponding integral gain and the robustness measure for

each PID setting. From the table we observe that only the controllers tuned by R-L, H-C, IMC

and the proposed methods are robust enough. Now a good compromise between performance

13

Table 3: PID settings for example 3

Kp Ti Td Ki εm

De Poar and O’Malley (P-M) [10] 1.459 2.667 0.25 0.547 10.46Rotstein and Lewin (R-L) [11] 2.25 5.76 0.2 0.391 4.01

Poulin and Pomerleau (P-P) [12] 2.025 4.738 0 0.427 9.04Huang and Chen (H-C) [13] 2.636 5.673 0.118 0.465 5.70

Tan et.al. [15] 2.428 2.381 0.098 1.02 7.06IMC [16] 2.634 2.52 0.154 1.045 5.29

ITSE-setpoint [17] 3.148 1.806 0.21 1.743 9.76Proposed 2.467 4.08 0.15 0.604 4.64

and robustness requires that the robustness measure for the unstable process lies between 4 and

6 compared with between 3 and 5 for the stable processes. The IMC method has the best load

rejection, as shown by the closed-loop system responses in Fig. 4(a) for a step setpoint change of

magnitude 1 at t = 0 following a setp load disturbance of magnitude 1 at t = 20 for the nominal

model. The PID controller by the H-C method was shown to be worse than that by the IMC

method in performance and robustness [16], so its response is omitted here. The PID controller

tuned by the R-L method has the best robustness, as shown by the closed-loop system responses

in Fig. 4(b) for a perturbed model with the deadtime increased by 20%. The new setting by the

proposed method has the best compromise between performance and robustness.

3.4 A multivariable process

The criteria can also be used to compare PID settings for multivariable processes. To illustrate,

consider the distillation column model reported by Wood and Berry [33]:

G(s) =

⎡⎢⎣

12.8e−s

16.7s+1−18.9e−3s

21s+1

6.6e−7s

10.9s+1−19.4e−3s

14.4s+1

⎤⎥⎦ (21)

14

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

Time

Pro

cess

Out

put

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

Time

Pro

cess

Out

put

(a) Nominal (b) Deadtime increases by 20%

Figure 4: Responses of different PID settings for example 3(solid: proposed; dashed: R-L; dashdotted: IMC)

The process is highly coupled and attracts much attention in the literature.

Table 4 shows the PID settings tuned by various methods found in the literature, and the PID

setting designed by solving the optimization problem proposed at the end of the previous section

with γm = 4. It is clear that the PID controllers given in [23, 33] have very large robustness

measures, and those given in [18, 22] have too small integral actions.

For the rest settings, the proposed PID has the best disturbance ejection, which can be shown

in Fig. 5. To test the robust performance of the controllers, suppose the process delays change,

and the perturbed model becomes

Gp(s) =

⎡⎢⎣

12.8e−2s

1+16.7s−18.9e−4s

1+21s

6.6e−10s

1+10.9s−19.4e−4s

1+14.4s

⎤⎥⎦ (22)

The disturbance responses for all the controllers are shown in Fig. 6. Again, the new setting by

the proposed method has the best compromise between performance and robustness.

15

Table 4: PID controller settings for example 4

PID controller σ(Ki) εm

[21]

[0.184+ 0.0469

s −0.0102− 0.0229s +0.0082s

−0.0674+ 0.0159s −0.537s −0.0660− 0.0155

s

]0.0111 2.647

[22]

[0.2833+ 0.0285

s −(0.04105+ 0.02185s )

0.09154+ 0.00971s −(0.121+ 0.0148

s )

]0.0056 3.927

[18]

[0.375(1+ 1

8.29s) 00 −0.075(1+ 1

23.6s)

]0.0032 4.130

[20]

[0.183(1+ 1

10.7s) 00 −0.072(1+ 1

10.7s)

]0.0111 4.152

[33]

[0.2(1+ 1

4.44s) 00 −0.004(1+ 1

2.67s)

]0.0150 7.551

[23]

[0.637(1+ 1

3.84s) 00 −0.096(1+ 1

7.40s)

]0.0130 8.756

[31]

[0.2796+ 0.0334

s −(0.0085s +0.0981s)

−(0.0381+ 0.0001s + 0.4913s

5s+1 ) −(0.1089+ 0.0136s )

]0.0131 3.988

0 10 20 30 40 50 60 70 80−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time

Pro

cess

Out

put

0 10 20 30 40 50 60 70 80−6

−5

−4

−3

−2

−1

0

1

Time

Pro

cess

Out

put

(a) Input disturbance at 1st channel (b) Input disturbance at 2nd channel

Figure 5: Process disturbance responses for example 4: nominal case(solid: proposed; dashdotted: [21]; dashed: [20])

16

0 10 20 30 40 50 60 70 80−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time

Pro

cess

Out

put

0 10 20 30 40 50 60 70 80−7

−6

−5

−4

−3

−2

−1

0

1

2

3

TimeP

roce

ss O

utpu

t

(a) Input disturbance at 1st channel (b) Input disturbance at 2nd channel

Figure 6: Process disturbance responses for example 4: perturbed case(solid: proposed; dashdotted: [21]; dashed: [20])

4 Conclusions

Criteria based on disturbance rejection and system robustness were proposed to assess the perfor-

mance of PID controllers. The robustness is measured by a two-block structured singular value,

and the disturbance rejection is measured by the minimum singular value of the integral gain

matrix. Examples showed that the criteria can be applied to a variety of processes, whether they

are stable, integrating or unstable; single-loop or multi-loop. It was also observed that robustness

measure should lie between 3 and 5 to have a better compromise on performance and robustness

for stable processes, and between 4 and 6 for unstable and integrating processes.

17

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19

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Biographies

Wen Tan received his B.Sc. degree in applied mathematics and M.Sc. degree in systems sci-

ence from the Xiamen University, China, and Ph.D. degree in automation from the South China

University of Technology, China, in 1990, 1993, and 1996, respectively. From October 1994

to February 1996, he was a Research Assistant with the Department of Mechanical Engineering

and Electronic Engineering, Hong Kong Polytechnic University, Hong Kong. After June 1996,

he joined the faculty of the Power Engineering Department at the North China Electric Power

University, China, where he was a Lecturer until December 1998 and an Associate Professor

from January 1999. From January 2000 to December 2001, he was a Postdoctoral Fellow in the

Department of Electrical and Computer Engineering at the University of Alberta, Canada. He is

currently a Professor with the Automation Department of the North China Electric Power Uni-

versity (Beijing), China. His research interests include robust and H∞ control with applications

in industrial processes.

Horacio J. Marquez

Tongwen Chen

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